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Why do integers under addition form an Abelian group?
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modern algebra
A simple algebraic curve, showing the graph of the algebraic equation y2 = x3 + 1.
In modern algebra, the set of integers under addition forms an Abelian group because it satisfies the four fundamental group axioms while also adhering to the commutative law.1
To qualify as an Abelian group, the system must meet these criteria:
- Closure: The sum of any two integers is always another integer.12
- Associativity: For any integers a, b, and c, the equation a + (b + c) = (a + b) + c holds true.12
- Identity Element: There exists an additive identity, 0, such that a + 0 = a for any integer a.132
- Inverse Element: For every integer a, there exists an inverse element, -a, such that a + (-a) = 0.12
- Commutativity: The group is defined as "Abelian" because it satisfies the commutative law, meaning a + b = b + a for all integers.124
Because the set of integers under addition satisfies all these properties, it is classified as an Abelian group.1